Self-oscillating mode for frequency modulation noncontact atomic force microscopy Franz J. Giessibl and Marco Tortonese Citation: Applied Physics Letters

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Self-oscillating mode for frequency modulation noncontact atomic force microscopy

Franz J. Giessibl and Marco Tortonese

Citation: Applied Physics Letters 70, 2529 (1997); doi: 10.1063/1.118910 View online: http://dx.doi.org/10.1063/1.118910

View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/70/19?ver=pdfcov Published by the AIP Publishing

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Self-oscillating mode for frequency modulation noncontact atomic force microscopy

Franz J. Giessibl

Universita¨t Augsburg, EKM, 86135 Augsburg, Germany Marco Tortonese

Park Scientific Instruments, Sunnyvale, California, 94089

~Received 21 January 1997; accepted for publication 12 March 1997!

Frequency modulation atomic force microscopy ~FM-AFM! has made imaging of surfaces in ultrahigh vacuum with atomic resolution possible. Here, we demonstrate a new approach which simplifies the implementation of FM-AFM considerably and enhances force sensitivity by directly exciting the cantilever with the thermal effects involved in the deflection measurement process. This approach reduces the mechanically oscillating mass by 6 to 8 orders of magnitude as compared to conventional FM-AFM, because external actuators and oscillating cantilever mounts are not needed.

Avoiding external actuators allows the use of cantilevers with very high oscillation frequencies, which results in improved force sensitivity. Further, the implementation and operation of this new technique is significantly simplified, because external actuator, bandpass filter, and phase shifter are eliminated. © 1997 American Institute of Physics. @S0003-6951~97!01419-8#

Resolving the surface structure of Si (111)-(737) by atomic force microscopy~AFM!¹has been awaited²since the conception of AFM in 1985. Frequency modulation AFM

~FM-AFM!³ has provided unprecedented resolution in vacuum⁴ and finally allowed atomic resolution of Si (111)-(737) in 1994.^5,6 In the past two years, several groups have performed refined studies on silicon^7,8and demonstrated atomic resolution on other semiconductors⁹ and insulators^10,11by using FM-AFM. The heart of an FM-AFM is a cantilever with positive feedback: the output of the deflection signal is fed back through an automatic gain control

~AGC!and a phase shifter to an actuator which excites the cantilever ~Fig. 1!. When the phase shift between the elon- gation of actuator and cantilever is adjusted to 1p^{/2, the} oscillation amplitude of the cantilever is Q times the ampli- tude of the actuator~Q is the mechanical quality factor, rang- ing from 10⁴ to 10⁵ in vacuum!. For cantilevers with an eigenfrequency n0 below the lowest eigenfrequency of the mount-actuator assembly, driving the cantilever with an external actuator is efficient and reliable. However, it is desir- able to use cantilevers with a very high n0 since the force resolution of an AFM improves proportional to the square root of n0 @Eq. ~19! in Ref. 3#. Most implementations of conventional FM-AFM are a compromise between the sensitivity gain achieved by using cantilevers with n0 higher than the lowest mechanical resonance of the mount-actuator assembly and the problems associated with operating the actuator above its lowest mechanical eigenfrequency. Usually, a bandpass filter is inserted into the feedback loop in order to ensure that the system oscillates at n0.³ Still, adjusting the phase shift can be tricky because operating the feedback loop beyond the eigenfrequency of the mechanical mount adds phase shifts which depend on the microscopic mechanical contact between cantilever and mount. Here we present a novel approach which utilizes an internal excitation process.

Thereby, many of the problems in conventional FM-AFM are overcome and cantilevers with much higher eigenfre- quencies can be used.

The experimental setup used in the present study con- sists of a piezoresistive cantilever¹² @~PL! Fig. 2# at room

temperature in a moderate vacuum ( p510²⁴ Pa). The over- all length of the PL used is L5175mm, the length of the legs is L575mm, the width w58mm. The PL has three layers: a SiO₂ layer with a thickness of tSiO₂50.1m^{m, a} doped layer withtdl51.0mm, and a pure silicon layer with tSi51.0mm, resulting in a total thickness oftPL52.1m^m.

A deflection zalters the resistance R_PL of the PL according to R_PL52 kV(11Sz). The sensitivity S of the PL used is (0.460.1)10²⁵ nm²¹ at room temperature. The PL is used as one of the four resistors in a Wheatstone bridge, the re- sistance of the three remaining resistors is R52.00 kV ~Fig.

3!. The output of the bridge is amplified by an instrumenta- tion amplifier with a gain of G5100. For small deflections,

V_out520.25GV₀Sz^. ~1!

The bridge bias is modulated according to

V₀5V_dc1V_modcos~2pn^t!,uV_modu!uV_dcu. ~2! The modulation of V₀ results in a deflectionzaccording to

FIG. 1. Conventional force sensor of a FM noncontact AFM. When the oscillating tip approaches the sample, the force between tip and sample causes a frequency shift. By adjusting the distance such that the frequency shift stays constant while the cantilever is scanned across a surface, a topo- graphic image is created. In the conventional sensor, the actuator and mount assembly oscillates at n0. Typically, the mass of the cantilever is 3 310²⁸times smaller than the mass of the whole sensor assembly.

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z~t!5zmod~n!cos~2pn^t1f!. ~3! Whenn is tuned, V_outshows a strong resonance forn5n0. The magnitude and width of the resonance are determined by the Q factor of the PL, which is Q5(460.4)10⁴. Q is ex- perimentally determined by the ratio betweenn0 and the frequency spread d ^{at which} zmod(n02d^/2)5zmod(n01d^/2)

51/&zmod(n0). At resonance, the phase shift between

V_outand V_modis 0, i.e., the phase shiftf ^betweenz^{(t) and} V_modis2p. The fact thatfis independent of the type of PL and the ultrasound-transmission properties of the mounting assembly is a major advantage over conventional FM-AFM implementations since an adjustable phase shifter is no longer needed.¹³

The experimental data shown in Fig. 4 give evidence of a linear dependence of zmod(n0) versus the time dependent component P_modof the electrical power P@P5V₀²/(4R_PL)# which is dissipated in the PL. P is separated in a constant part P_dc5V_dc²/(4R_PL) and P_mod(t) 52V_dcV_modcos(2pn^t)/(4RPL)for uV_modu!uV_dcu. The spatial and temporal distribution of T(x,y ,z,t) is the difference be- tween the local temperature of the PL and ambient tempera-

ture and is described by the heat diffusion equation:¹⁴ ]^T~x, y ,z,t!

]^t ⁵ k

r^c ^D^T^~^{x,y ,z,t}^!1 1

r^c ^p^~^{x,y ,z,t}^!^, ^~⁴^! wherekis the heat conductivity,ris the mass density, c the specific heat ~see Table I! and p(x, y ,z,t) the local heating power density. Solving Eq. ~4! in three spatial dimensions for the actual geometry of the PL~see Fig. 2!is tedious and may only be done numerically. Reducing the problem to one spatial dimension yields analytical solutions which gain in- sight into the physics of the self-oscillation mode. For the PLs, the heating power density is given by P divided by the

‘‘active volume’’ V*, i.e., the volume in which heat is generated. The resistance of the top part of the PL ~section de- fined by x.Lin Fig. 2!can be neglected. Therefore, heat is only generated in the legs of the PL and V*₅2LtPLw ~section defined by 0,x,L in Fig. 2!. Equation ~4! does not account for heat radiation, which is negligible ~typically, heat radiation is less than 0.25% of heat conduction!. T(x,t) is split into a stationary and a dynamic component:

T~x,t!5T_dc~x!1T_mod~x,t!. ~5! Choosing appropriate boundary conditions and integration of Eq. ~4!yields

T_dc~x!5T_max~2x/L2x²/L²! with T_max

5P_dc/V*_L²/2k^. ~6! For the calculation of the time-dependent temperature com- ponent T_mod(x,t), the heat conduction contribution in Eq.~4!

FIG. 2. Composition of a PL and temperature profile during operation.

FIG. 3. Schematic of the self-oscillating force sensor utilizing a PL.

FIG. 4. Deflection amplitudezmod(n0) of the PL as a function of Pmod.

TABLE I. Mechanical and thermal properties^a,b,cof Si and SiO2 dioxide

~300 K!.

E@10¹¹N/m²# r@kg/m³# a@10²⁶/K# k@W/mK# c@J/kg K#

Si 1.7 2329 2.33 156 690

SiO₂ 0.79 2500 0.55 1.4 170

aLandolt–Boernstein, Numerical Data and Functional Relationships in Sci- ence and Technology, edited by O. Madelung, M. Schultz, and H. Weiss, New Series~Springer, Berlin, 1982!, Vol. 17a.

bK. E. Petersen, Proc. IEEE 70, 420~1982!.

cH. Kuchling, Taschenbuch der Physik,~Harri Deutsch, Thun und Frankfurt/

Main, 1982!, p. 596.

2530 Appl. Phys. Lett., Vol. 70, No. 19, 12 May 1997 F. J. Giessibl and M. Tortonese

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is neglected~will be justified below!. Equation~4!reduces to ]^T~x,t!

]^t ⁵ 1

r^cV* ^P^mod^~^t^! ^{for 0}^,^x^,^L^. ^~⁷^! For 0,x,L integration of Eq.~7!yields

T_mod~x,t!5P_mod~t2C/4!

2pn0r^cV* ⁵^:T^mod,leg^~^t^! ^~⁸^! with C:51/n0. Outside the legs of the PL:

]^Tmod~x,t! ]^t ⁵

k r^c

]²^Tmod~x,t!

]^x² ^{for x}^,⁰^~^x^.^L^. ^~⁹^! The ansatz T:5T₀Re$^eⁱ⁽²^pn^l²^kx)%yields a solution for Eq.~9!: an exponentially decaying temperature distribution

T_mod~x,t!5T₀e^x/^lcos~2pn0t1x/l! for x,0 ~10a! and for x.L

T_mod~x,t!5T₀e^2~^x^2L!^/^l cos~2pn0t2~x2L!/l! ~10b! with T₀5T_mod,leg~t50! and l5

A

_pn^k0r^c^. ^~¹¹^! For the PL used ~n0543.0 kHz), the thermal penetration depth l (lSi527mm) is still considerably smaller than the length of the legs (L575m^{m). Since}lis much larger than v ^andtPL, the one-dimensional approximation of Eq.~4! is justified for x:0,x,L. Outside the legs of the PL~Fig. 2!, the cross section of the PLs increases in the y and z direction for x,0 and in y direction for x.L and T_mod(x,t) outside of V* drops at a rate given by'e^2r^/^l, whereris the dis- tance to V*. It remains to be shown that the heat conduction term in Eq.~8!can be neglected indeed. Using Eq.~10a!, the thermal power which leaks out of the legs into the substrate of the PL is given by

P_leak5] ]^t

E

0

2`

cr^wtPLT_mod~x,t!dx

5@2pn0lcr^wtPLT_mod,leg~t2C/8!/#2&. ~12! Comparing P_leak to P_modyields:

uP_leak/ P_modu5l/4&L!1 . ~13!

Figure 2 shows the temperature profile along the PL. Be- cause of a ‘‘bimetallic’’ effect, T_mod,leg(t) causes an excitation amplitude of the PL given by¹⁵

zexc~t!56tSiO₂E_SiO

2

tPL

2 E_Si ~aSi2aSiO₂!L~L2L/2!T_mod,leg~t! 51.27 nm K²¹T_mod,leg~t!, ~14! where E and a are Young’s moduli and linear thermal ex- pansion coefficients of Si and SiO₂, respectively ~Table I!. At resonance, this excitation causes the PL to oscillate by a factor of Q ~and phase shifted by2p^/2!times the excitation amplitude. Combining Eqs.~8!and~14!yields

z~t!523Q P_mod~t! pn0r^cV*

tSiO₂E_SiO

2

tPL 2 E_Si

3~aSi2aSiO₂!L~L2L/2!. ~15!

Figure 4 shows the dependence of zmod(n0) as a function of P_mod. Considering the large number of factors ~materials properties and geometric dimensions!embedded in Eq.~15!, the agreement between theory and experiment is excellent. It is noted, though, that the result according to Eq.~15!is sys- tematically smaller than the experimental data. The data in Fig. 4 suggest that the actual product of Q and S is'35%

larger than the value determined by their standard measuring procedure. Since the inaccuracies of the Q @Eq.~15!#and S

@Eq. ~1!#measurements are 10% and 25%, respectively, the deviation is fully within the expected accuracy.

In summary, we have demonstrated a novel technique for the excitation of the cantilever which can be used for the following:

~1! Significant simplification of FM-AFM, since external actuator, phase shifter, and bandpass filter are eliminated.

~2! Improved force resolution in conventional FM-AFM.

~3! Improved mass resolution of micromechanical calo- rimeters.¹⁶

~4! Operation of multiple cantilevers on a single chip in FM- AFM mode.¹⁷

Also, we have developed a model which explains the operation of self-excitation with excellent agreement to the experiment and provides a framework for future improved force detectors.

The authors thank T. R. Albrecht, J. Alexander, J. Mann- hart, and C. Masser for helpful discussions. This work was partially supported by BMBF Grant No. 13N6918.

1G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930~1986!.

2Atomic resolution of Si(111)-(737) by AFM was considered a mile- stone in the progress of AFM, because the atomic resolution capability of scanning tunneling microscopy~STM!was recognized in the scientific community after the surface structure of Si(111)-(737) had been re- vealed by STM.

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~1993!.

13We have operated a variety of PLs withn0ranging from 43 to 207 kHz and foundf52p60.1 for all of them.

14R. P. Feynman, B. Leighton, and M. Sands, The Feynman Lectures on Physics, II-3-4, Reading Massachusetts, 1963.

15Calculated by integration of the differential equation for the bending curve: ]²^z/]^x²52M (x)/ESiI, M (x)5ESiO₂(aSi2aSiO₂)tSiO₂wtPL/2 for tSiO₂!tPLand I5wtPL

3 /12.

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Appl. Phys. Lett., Vol. 70, No. 19, 12 May 1997 F. J. Giessibl and M. Tortonese